That's usually what I want but I tend to drill and use a bolt extractor. Left handed bit is a good idea.
I usually start drilling with a left handed bit, if it doesn't grab and pull the broken bolt out quickly I switch to right handed. Work provides unlimited right hand drill bits and doesn't provide left handed drill bits. If I'm going to destroy a bit it might as well be one work provides. I have 5 different styles of extractors in my box though try not to have to use them. I try and get lucky working it out with a pick if it didn't break because of rust. If it doesn't come out easy and I can weld a nut on I weld a nut on I go that route. If not it's left handed bit, open up to put a smallish extractor in and see if it moves, if it's stubborn keep drilling till I can put a larger extractor in and hope for the best. The younger guys at work go straight to the ez outs when welding a nut on is the better option most times.
I don't have a dedicated stubby drill bit set though will cut down a normal bit if necessary. There's a couple stubby left handed drill bit sets out there, one is on my buy it one of these days lists.
I have put a 1/4" drill bit in a right angle die grinder, speed control is an issue keeping it spinning slow. I used a pair of vise grips as a makeshift regulator though it doesn't work great.
